2. Knot colouring, labelling arcs with integers, colouring mod n, splittable links, Borromean rings, chess board structure on a diagram, reduced connected diagram determines quadrilateral decomposition, crossing equations sum to 0.

3. A systematic approach to colouring, colouring through the determinant of a link, n is a colouring number if and only if n and the determination of the link have a common factor, the colouring group Col(K) is stronger than det(K).

4. Mirrors and Knot coding, mirrors, reversing orientation, chiral knots, neither colouring nor determinants detect chirality.

5. Alexander polynomials, defined as the determinant of an arc labelling, effects of mirrors and reverses, vanishing on a splittable link, knot sums , reef and granny knots, behaviour of Alexander polynomial under sum of knots.

6. Bridge number plats and braids, generators of the braid groups, Markov moves, Seifert circles, braid index is the same as the Seifert index (Vogel's proof). 2-bridge knots. Schubert normal form. Daisy chains are 4-plats. Simplifying a 4-plat diagram

7. The Jones polynomial, bracket polynomial, state sums, skein relations, Conway knot and Kinoshita-Terasaka knot.

8.Alternating Links, highest degree term related to black white regions and writhe of knot, connected reduced alternating diagram then span Jones=number crossings.

9.Tangles, algebra of tangles, relation to daisy chains, continued fraction description of determinant, application to enzyme action on DNA.

10. Genus and knot sum, genus of oriented surfaces, Seifert surfaces, genus related to number of crossings and number of Seifert circles, genus is additive, corollary: knots do not have inverses.

11. The Conway polynomial, description via skein relations, ease of calculation, effect of mirrors and orientation change, meaning of coefficients, relation with Alexander polynomial, HOMFLY polynomial generalises Conway and Jones.

Brian Sanderson, March 1999